If the objective function $Z=px+qy,p>0,q>0$ of a linear programming problem attains its optimal value at the points (4, 7) and (5, 5) and $pq = 50$ then

$p = 10,q=5$

The correct answer is Option (2) → $p = 10,q=5$

Optimal value at both points means:

$4p + 7q = 5p + 5q$

$\Rightarrow 7q - 5q = 5p - 4p$

$\Rightarrow 2q = p$

Given: $pq = 50$

$\Rightarrow (2q)\,q = 50$

$\Rightarrow 2q^{2} = 50$

$\Rightarrow q^{2} = 25$

$\Rightarrow q = 5$ (since $q>0$)

$p = 2q = 10$